INDUCTION

INDUCTION, in logic, is that process of the understanding by which, from a number of particular truths perceived by simple apprehension, and diligently compared together, we infer another truth which is always general and sometimes universal. It is perhaps needless to observe, that in the process of induction the truths to be compared must be of the same kind, or relate to objects having a similar nature; for the merest tyro in

A science

Induction. science knows that physical truths cannot be compared with moral truths, nor the truths of pure mathematics with either.

That the method of induction is a just logic, has been sufficiently evinced elsewhere (see LOGIC, Part III. chap. V. and PHILOSOPHY, n^o 73—78. Encycl.), and is now indeed generally admitted. It is even admitted by British philosophers to be the only method of reasoning by which any progress can be made in the physical sciences; for the laws of Nature can be discovered only by accurate experiments, and by carefully noting the agreements and the differences, however minute, which are thus found among the phenomena apparently similar. It is not, however, commonly said that induction is the method of reasoning employed by the mathematicians; and the writer of this article long thought, with others, that in pure geometry the reasoning is strictly syllogistical. Mature reflection, however, has

* Appendix to Vol. III. of Sketches of the History of Man. led him to doubt, with Doctor Reid*, the truth of the generally received opinion, to doubt even whether by categorical syllogisms any thing whatever can be proved.

To the idolaters of Aristotle we are perfectly aware that this will appear an extravagant paradox; but to the votaries of truth, we do not despair of making it very evident, that for such doubts there is some foundation.

We are led into this disquisition to counteract, in some degree, what we think the pernicious tendency of the philosophy of Kant, which attempts have been lately made to introduce into this country. Of this philosophy we shall endeavour to give something like a distinct view in the proper place. It is sufficient to observe here, that it rests upon the hypothesis, that "we are in possession of certain notions a priori, which are absolutely independent of all experience, although the objects of experience correspond with them; and which are distinguished by necessity and strict universality." These innate and universal notions, Kant considers as a set of categories, from which is to be deduced all such knowledge as deserves the name of science; and he talks, of course, or at least his English translators represent him talking, with great contempt, of inductive reasoning, and substituting syllogistic demonstration in its stead.

As his categories are not familiar to our readers, we shall, in this place, examine syllogisms connected with the categories of Aristotle, which are at least more intelligible than those of Kant, and which, being likewise general notions, must, in argument, be managed in the same way. Now the fundamental axiom upon which every categorical syllogism rests, is the well known proposition, which affirms, that "whatever may be predicated of a whole genus, may be predicated of every species and of every individual comprehended under that genus." This is indeed an undoubted truth; but it cannot constitute a foundation for reasoning from the genus to the species or the individual; because we cannot possibly know what can be predicated of the genus till we know what can be predicated of all the individuals ranged under it. Indeed it is only by ascertaining, through the medium of induction, what can be predicated, and what not, of a number of individuals, that we come to form such notions as those of genera and species; and therefore, in a syllogism strictly categorical, the propositions, which constitute the premises, and are taken for granted, are those alone which are capable of proof; whilst the conclusion, which the logician pre-

tends to demonstrate, must be evident to intuition or experience, otherwise the premises could not be known to be true. The analysis of a few syllogisms will make this apparent to every reader.

Dr Wallis, who, to an intimate acquaintance with the Aristotelian logic, added much mathematical and physical knowledge, gives the following syllogism as a perfect example of this mode of reasoning in the first figure, to which it is known that all the other figures may be reduced:—

Omne animal est sensu praditum.
Socrates est animal. Ergo
Socrates est sensu praditus.

Here the proposition to be demonstrated is, that Socrates is endowed with sense; and the propositions assumed as self-evident truths, upon which the demonstration is to be built, are, that "every animal is endowed with sense," and that "Socrates is an animal." But how comes the demonstrator to know that "every animal is endowed with sense?" To this question we are not aware of any answer which can be given, except this, that mankind have agreed to call every being, which they perceive to be endowed with sense, an animal. Let this, then, be supposed the true answer: the next question to be put to the demonstrator is, How he comes to know that Socrates is an animal? If we have answered the former question properly, or, in other words, if it be essential to this genus of beings to be endowed with sense, it is obvious that he can know that Socrates is an animal only by perceiving him to be endowed with sense; and therefore, in this syllogism, the proposition to be proved is the very first of the three of which the truth is perceived; and it is perceived intuitively, and not inferred from others by a process of reasoning.

Though there are ten categories and five predicates, there are but two kinds of categorical propositions, viz. Those in which the property or accident is predicated of the substance to which it belongs, and those in which the genus is predicated of the species or individual. Of the former kind is the proposition pretended to be proved by the syllogism which we have considered; of the latter, is that which is proved by the following:

Quicquid sensu praditum, est animal.
Socrates est sensu praditus. Ergo
Socrates est animal.

That this is a categorical syllogism, legitimate in mode and figure, will be denied by no man who is not an absolute stranger to the very first principles of the Aristotelian logic; but it requires little attention indeed to perceive that it proves nothing. The imposition of names is a thing so perfectly arbitrary, that the being, or class of beings, which in Latin and English is called animal, is with equal propriety in Greek called ζῷον, and in Hebrew חַי. To a native of Greece, therefore, and to an ancient Hebrew, the major proposition of this syllogism would have been wholly unintelligible; but had either of those persons been told by a man of known veracity, and acquainted with the Latin tongue, that every thing endowed with sense was, by the Romans, called animal, he would then have understood the proposition, admitted its truth without hesitation, and have henceforth

Induction. henceforth known that Socrates and Moses, and every thing else which he perceived to be endowed with sense, would at Rome be called animal. This knowledge, however, would not have rested upon demonstrative reasoning of any kind, but upon the credibility of his informer, and the intuitive evidence of his own senses.

It will perhaps be said, that the two syllogisms which we have examined are improper examples, because the truth to be proved by the former is self-evident, whilst that which is meant to be established by the latter is merely verbal, and therefore arbitrary. But the following is liable to neither of these objections:

All animals are mortal.

Man is an animal; therefore

Man is mortal.

Here it would be proper to ask the demonstrator, upon what grounds he so confidently pronounces all animals to be mortal? The proposition is so far from expressing a self-evident truth, that, previous to the entrance of sin and death into the world, the first man had surely no conception of mortality. He acquired the notion, however, by experience, when he saw the animals die in succession around him; and when he observed that no animal with which he was acquainted, not even his own son, escaped death, he would conclude that all animals, without exception, are mortal. This conclusion, however, could not be built upon syllogistic reasoning, nor yet upon intuition, but partly upon experience and partly on analogy. As far as his experience went, the proof, by induction, of the mortality of all animals was complete; but there are many animals in the ocean, and perhaps on the earth, which he never saw, and of whose mortality therefore he could affirm nothing but from analogy, i. e. from concluding, as the constitution of the human mind compels us to conclude, that Nature is uniform throughout the universe, and that similar causes, whether known or unknown, will, in similar circumstances, produce, at all times, similar effects. It is to be observed of this syllogism, as of the first which we have considered, that the proposition, which it pretends to demonstrate, is one of those truths known by experience, from which, by the process of induction, we infer the major of the premises to be true; and that therefore the reasoning, if reasoning it can be called, runs in a circle.

Yet by a concatenation of syllogisms have logicians pretended that a long series of important truths may be discovered and demonstrated; and even Wallis himself seems to think, that this is the instrument by which the mathematicians have deduced, from a few postulates, accurate definitions, and undeniable axioms, all the truths of their demonstrative science. Let us try the truth of this opinion by analysing some of Euclid's demonstrations.

In the short article PRINCIPLE (Encycl.), it has been shewn, that all our first truths are particular, and that it is by applying to them the rules of induction that we form general truths or axioms—even the axioms of pure geometry. As this science treats not of real external things, but merely of ideas or conceptions, the creatures of our minds, it is obvious, that its definitions may be perfectly accurate, the induction by which its axioms are formed complete, and therefore the axioms themselves universal propositions. The use of these axioms

is merely to shorten the different processes of geometrical reasoning, and not, as has sometimes been absurdly supposed, to be made the parents or causes of particular truths. No truth, whether general or particular, can, in any sense of the word, be the cause of another truth. If it were not true that all individual figures, of whatever form, comprehending a portion of space equal to a portion comprehended by any other individual figure, whether of the same form with some of them, or of a form different from them all, are equal to one another, it would not be true that "things in general, which are equal to the same thing, or that magnitudes which coincide, or exactly fill the same space," are respectively equal to one another; and therefore the first and eight of Euclid's axioms would be false. So far are these axioms, or general truths, from being the parents of particular truths, that, as conceived by us, they may, with greater propriety, be termed their offspring. They are indeed nothing more than general expressions, comprehending all particular truths of the same kind. When a mathematical proposition therefore is enounced, if the terms, of which it is composed, or the figures of which a certain relation is predicated, can be brought together and immediately compared, no demonstration is necessary to point out its truth or falsehood. It is indeed intuitively perceived to be either comprehended under, or contrary to some known axiom of the science; but it has the evidence of truth or falsehood in itself, and not in consequence of that axiom. When the figures or symbols cannot be immediately compared together, it is then, and only then, that recourse is had to demonstration; which proceeds, not in a series of syllogisms, but by a process of ideal mensuration or induction. A figure or symbol is conceived, which may be compared with each of the principal figures or symbols, or, if that cannot be, with one of them, and then another, which may be compared with it, till through a series of well known intermediate relations, a comparison is made between the terms of the original proposition, of which the truth or falsehood is then perceived.

Thus in the 47th proposition of the first book of Euclid's Elements, the author proposes to demonstrate the equality between the square of the hypotenuse of a right angled triangle, and the sum of the squares described on the other two sides; but he does not proceed in the way of categorical syllogisms, by raising his demonstration on some universal truth relating to the genus of squares. On the contrary, he proceeds to measure the three squares of which he has affirmed a certain relation; but as they cannot be immediately compared together, he directs the largest of them to be divided into two parallelograms, according to a rule which he had formerly ascertained to be just; and as these parallelograms can, as little as the square of which they are the constituent parts, be compared with the squares of the other two sides of the triangle, he thinks of some intermediate figure which may be applied as a common measure to the squares and the parallelograms. Accordingly, having before found that a parallelogram, or square, is exactly double of a triangle standing on the same base and between the same parallels with it, he constructs triangles upon the same base, and between the same parallels with his parallelograms, and the squares of the sides containing the right angle of the original triangle; and finding, by a process formerly shewn to be just,

Induction that the triangles on the bases of the parallelograms are precisely equal to the triangles on the bases of the squares, he perceives at once that the two parallelograms, of which the largest square is composed, must be equal to the sum of the two lesser squares; and the truth of the proposition is demonstrated.

In the course of this demonstration, there is not so much as one truth inferred from another by syllogism, but all are perceived in succession by a series of simple apprehensions. Euclid, indeed, after finding the triangle constructed on the base of one of the parallelograms to be equal to the triangle constructed on the base of one of the squares, introduces an axiom, and says, "but the doubles of equals are equal to one another; therefore the parallelogram is equal to the square." But if from this mode of expression any man conceive the axiom or universal truth to be the cause of the truth more particular, or suppose that the latter could not be apprehended without a previous knowledge of the former, he is a stranger to the nature of evidence, and to the process of generalization, by which axioms are formed.

If we examine the problems of this ancient geometry, we shall find that the truth of them is proved by the very same means which he makes use of to point out the truth of his theorems. Thus, the first problem of his immortal work is, "to describe an equilateral triangle on a given finite straight line;" and not only is this to be done, but the method by which it is done must be such as can be shown to be incontrovertibly just. The sides of a triangle, however, cannot be applied to each other so as to be immediately compared; for they are conceived to be immovable among themselves. A common measure, therefore, or something equivalent to a common measure, must be found, by which the triangle may be constructed, and the equality of its three sides afterwards evinced; and this equivalent Euclid finds in the circle.

By contemplating the properties of the circle, it was easy to perceive that all its radii must be equal to one another. He therefore directs two circles to be described from the opposite extremities of the given finite straight line, so as that it may be the radius of each of them; and from the point in which the circles intersect one another, he orders lines to be drawn to the extreme points of the given line, affirming that these three lines constitute an equilateral triangle. To convince his reader of the truth of this affirmation, he has only to put him in mind, that from the properties of the circle, the lines which he has drawn must be each equal to the given line, and of course all the three equal to one another; and this mutual equality is perceived by simple apprehension, and not inferred by syllogistic reasoning. Euclid, indeed, by introducing into the demonstration his first axiom, gives to it the form of a syllogism; but that syllogism proves nothing; for if the equality of the three sides of the triangle were not intuitively perceived in their position and the properties of the circle, the first axiom would itself be a falsehood. So true it is that categorical syllogisms have no place in geometrical reasoning; which is as strictly experimental and inductive as the reasoning employed in the various branches of physics.

But if this be so, how come the truths of pure geometry to be necessary, so that the contrary of any one

of them is clearly perceived to be impossible; whilst Induction physical truths are all contingent, so that there is not one of them of which the direct contrary may not easily be conceived?

That there is not one physical truth, of which the contrary may not be conceived, is not perhaps so certain as has generally been imagined; but admitting the fact to be as it has commonly been stated, the apparent difference between this class of truths and those of pure geometry, may be easily accounted for, without supposing that the former rests upon a kind of evidence totally different from that which supports the fabric of the latter.

The objects of pure geometry, as we have already observed, are the creatures of our own minds, which contain in them nothing concealed from our view. As the mathematician treats them merely as measurable quantities, he knows, with the utmost precision, upon what particular properties the relation affirmed to subsist between any two or more of them must absolutely depend; and he cannot possibly entertain a doubt but it will be found to have place among all quantities having the same properties, because it depends upon them, and upon them alone. His process of induction, therefore, by a series of ideal measurements, is always complete, and exhausts the subject; but in physical enquiries the case is widely different. The subjects which employ the physical enquirer are not his own ideas, and their various relations, but the properties, powers, and relations of the bodies which compose the universe; and of those bodies he knows neither the substance, internal structure, nor all the qualities: so that he can very seldom discover with certainty upon what particular property or properties the phenomena of the corporeal world, or the relations which subsist among different bodies, depend. He expects, indeed, with confidence, not inferior to that with which he admits a mathematical demonstration, that any corporeal phenomenon, which he has observed in certain circumstances, will be always observed in circumstances exactly similar; but the misfortune is, that he can very seldom be ascertained of this similarity. He does not know any one piece of matter as it is in itself; he cannot separate its various properties; and of course cannot attribute to any one property the effects or apparent effects which proceed exclusively from it. Indeed, the properties of bodies are so closely interwoven, that by human means they cannot be completely separated; and hence the most cautious investigator is apt to attribute to some one or two properties, an event which in reality results perhaps from many. (See PHILOSOPHY and PHYSICS, Encycl.) This the geometer never does. He knows perfectly that the relation of equality which subsists between the three angles of a plain triangle and two right angles, depends not upon the size of the triangles, the matter of which they are conceived to be made, the particular place which they occupy in the universe, or upon any one circumstance whatever besides their triangularity, and the angles of their corollaries being exactly right angles; and it is upon this power of discrimination which we have in the conceptions of pure geometry, and have not in the objects of physics, that the truths of the one science are perceived to be necessary, while those of the other appear to be contingent; though the mode of demonstration is the same

Inertia, same in both, or at least equally removed from categorical syllogisms.